OFFSET
0,52
COMMENTS
The reverse-alternating sum of a partition (y_1,...,y_k) is Sum_i (-1)^(k-i) y_i. This is equal to (-1)^(m-1) times the number of odd parts in the conjugate partition, where m is the number of parts. So T(n,k) is the number of strict integer partitions of n such that k is equal to (-1)^(m-1) times the number of odd conjugate parts.
By conjugation, T(n,k) is also equal to the number of integer partitions of n covering an initial interval of positive integers such that k is equal to (-1)^(r-1) times the number of odd parts, where r is the greatest part.
Also the number of reversed strict integer partitions of n with alternating sum k.
EXAMPLE
Triangle begins:
1
0 1
0 0 1
0 1 0 1
0 1 0 0 1
0 1 1 0 0 1
0 1 1 0 1 0 1
0 1 1 1 0 1 0 1
0 1 1 1 0 1 1 0 1
0 1 1 1 1 0 2 1 0 1
0 1 1 1 2 0 1 2 1 0 1
0 1 1 1 2 1 0 2 2 1 0 1
0 1 1 1 2 2 0 1 3 2 1 0 1
0 1 1 1 2 3 1 0 2 3 2 1 0 1
0 1 1 1 2 3 3 0 1 3 3 2 1 0 1
0 1 1 1 2 3 4 1 0 3 4 3 2 1 0 1
0 1 1 1 2 3 5 3 0 1 4 4 3 2 1 0 1
0 1 1 1 2 3 5 5 1 0 3 5 4 3 2 1 0 1
0 1 1 1 2 3 5 6 4 0 1 5 6 4 3 2 1 0 1
For example, the partitions counted by row n = 15 are (empty columns shown as dots, A...F = 10..15):
. E1 D2 C3 B4 A5 96 87 . 762 654 843 A32 C21 . F
9321 7431 6432 861 753 942 B31
8421 6531 54321 852 A41
7521 951
MATHEMATICA
sats[y_]:=Sum[(-1)^(i-Length[y])*y[[i]], {i, Length[y]}];
Table[Length[Select[IntegerPartitions[n], UnsameQ@@#&&sats[#]==k&]], {n, 0, 12}, {k, -n, n, 2}]
CROSSREFS
Row sums are A000009.
The non-strict version is A344612.
The right halves of even-indexed rows are A344649.
The non-reverse non-strict version is the right half of A344651, which is A239830 interleaved with A239829.
A344610 counts partitions of n by positive reverse-alternating sum.
A344611 counts partitions of 2n with reverse-alternating sum >= 0.
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jun 05 2021
STATUS
approved